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Controlling lattice complexity or inhomogeneity of the doped oxidesuperconductors is shown to be a key term for control of the superconductivitycritical temperature. All high temperature superconductors show anisotropicgaps and/or multi-gap superconductivity in the clean limit. They show the shaperesonance in superconducting gaps that is a type of Fano resonance driven bythe Josephson-like term for pairs transfer. These resonances occur where one ofthe multiple Fermi surface spots is near a 2.5 Lifshitz transition. The Fanoantiresonance occurs between a BCS condensate and a BEC-like condensate. In asystem near a 2.5 Lifshitz transition the lattice is close to an instability.This drives to the formation of a granular superconducting phase made of acharge density metal formed by puddles of ordered oxygen interstitials orordered local lattice distortions (static short range charge density waves)detected by scanning nano-x-ray diffraction imaging. We show that the scalefree networks of superconducting grains tuned to a Fano resonance is anessential feature of high temperature superconductors that favours high Tc.

Scale free networks of superconducting striped grains tuned at Fano resonances for high Tc